To find the probability of John not winning the prize after three chances of throwing at the target, we first need to determine the probability of missing the target in a single throw.
Since the probability of hitting the target is 70%, the probability of missing the target is:
\[ P(\text{missing}) = 1 - P(\text{hitting}) = 1 - 0.70 = 0.30 \]
Now, we need to find the probability that John misses the target on all three of his throws. Since the throws are independent events, we can multiply the probabilities of missing the target for each throw:
\[ P(\text{missing all 3 throws}) = P(\text{missing})^3 = (0.30)^3 \]
Calculating \( (0.30)^3 \):
\[ 0.30^3 = 0.30 \times 0.30 \times 0.30 = 0.027 \]
Thus, the probability of John not winning the prize (i.e., missing the target on all three throws) is:
\[ \boxed{0.027} \]